Fig. 1: Carbon spin manipulation with and without magnetic field.

a Schematic structure of a nitrogen-vacancy center in diamond, where carbon (C) adjacent to a vacancy (e) is replaced by impurity nitrogen (N), whose spin error is protected by two carbon isotope spins serving as ancilla qubits for quantum error correction. The green-colored ¹³C atoms’ placement is just an artistic choice. b Quantization axis of carbon nuclear spins, which is determined by the hyperfine field by the electron spin as well as the external magnetic field if applied. c Simulated Rabi frequency dependence of the fidelity (F) of the holonomic controlled-phase gate based on the geometric phase between two carbon nuclear spins (¹³C₁, 1.14 MHz; ¹³C₂, 0.33 MHz) under a zero magnetic field (blue line), 100 Gauss (green line), 1000 gauss (red line), and 10000 Gauss (orange line). The angle between the external and hyperfine magnetic fields is set at 45 degrees for both carbon nuclear spins. Under a magnetic field, the fidelity is limited by the influence of the two quantum axes, but under a zero magnetic field, the fidelity improves by decreasing the Rabi frequency. The nitrogen nuclear spin is not considered here because the quantization axis does not change with the presence of a magnetic field. d Energy level diagram of the relevant spin system containing one nitrogen and two carbon isotopes together with an optically detected magnetic resonance spectrum relating to the |0⟩_s−|±1⟩_s transition. PL denotes photoluminescence. \({D}_{0}=2.877\) MHz is the zero-field splitting. Hyperfine splittings are observed (¹⁴N,2.2 MHz;¹³C₁,1.14 MHz;¹³C₂,0.33 MHz). The energy levels inside the blue rectangles are used in this study.